Squaring mechanism



Oct. 18, 1949. L, w, [MM 2,485,200

SQUARING MECHANISM Filed Oct. 29, 1943 Q 2 Sheets-Sheet 1 INVENTOR. Leel/1's W I m m Oct. 18, 1949. Ljw. lMM

' SQUARING MECHANISM 2 Sheets-Sheet 2 Filed Oct. 29, 1945 INVENTOR.Lea/11$ WIzzzm A TTORNEy.

Patented Oct. 18, 1949 SQUARING MECHANISM Lewis William Imm, Glendale,Calif., .assignor to Librascope, Incorporated, Burbank, Calif., acorporation of California Application October 29, 1943, Serial No.508,231

13 Claims. 1

The object of this invention is to provide a squaring mechanism of highaccuracy, by which a number may be directly squared or conversely thesquare root of a number may .be obtained.

Another object of the invention is to provide a squaring mechanism ofextreme simplicity in design and which may be cheaply manufactured.

Another object of the invention is to provide a squaring mechanism of adesign which is extremely rugged in character, and which may be easilymanufactured in quantities.

One of the essential features of the invention is a squaring mechanismincorporating a cone and cylinder with a cable attached, to be unwoundfrom either the cone or cylinder and wound on the other. The term coneas used herein is not confined to a mathematical cone or a solidgenerated by the rotation of a right triangle about one of .its legs asan axis but is used in its broader sense to include cone shapedstructures such as a frustrumof a cone and irrespective of whether it issolider hollow.

Another object of the invention is to provide two cables interconnectingthe cone and cylinder so that one of the cables is wound on the cylinderor cone, while the other cable is being unwound from the same, therebyeliminating the necessity of springs to actuate either the cone or thecylinder.

Obviously it is impractical for the cone to taper down to an absolutezero value. In practice the diameter of the small end of the cone mustbe some fixed value above zero such as, for instance,

while the diameter of the large end of the cone would have a greaterdiameter such as, for instance, 1 The average diameter therefore of thecone would be, in the instance given, 1", and preferably the diameter ofthe cylinder would be the same value as the mean diameter of the cone.It wouldat first appear that it would be impossible to square a numberso small as to fall Within the range of that'part of the theoreticalcone between a zero value and the value of the small part of the actualcone. One of the objects of this invention is to provide means wherebythese small numbers may be squared by utilizing the actual cone andcylinder and in eiiect transferring the zero part of the theoreticalcone to apart of the actual cone such as "its small end or even its mid'portion. This can be accomplished by adifierential mechanisminterposedbetween the cylinder and the cone and actuated by both ofthem, which differential mechanism would not be necessary in case thecone could 'be constructed as a theoretical cone, tapering down to anactual zero value.

Let it be assumed that we desire to square the number a, which may be avariablenumher, and let assume that the radius of the cone at itssmallest end is c. The square of :v+c r +2rc+e The object of thedifferential mechanism is to eliminate the 2x0, and the scale employedin conjunction with the output can have its zero value at the point 0This leaves only which is the square of the number sought to be squared.

The invention may be better understood by referring to the attacheddrawings in which Fig. l is a plan view of my improved squaring"mechanism.

Fig. 2 is a frontelevational view thereof.

Fig. 3 is a elevationalview of fragments of the cylinder and cone, withthe interconnecting cables andwi th certain of the threads on the conebeing viewed through magnifying glasses.

Fig. 4"is a side elevational view of the mechanism shown in Fig. 1.

Figs. 5 and 6 are diagrammatic views to illustrate the differentialtapers to compensate for slack in the cables.

Fig. 7 is an exploded view of the differential mechanism.

Fig. '8 is aside eievatlonal-view of the differential mechanism talcennnthe line 8-4! of Fig. =9.

Fig. il -is across sectional view taken on the line 99 of Fig. 8.

Fig. 1-0 is a cross sectional view taken on the line ML-I0 of Fig. 8.

Fig. l-lis aside elevational view of a portion of the differential-1'neehan-ism taken on the line =|-l--H of Fig. '10.

Fig. "12 is a side elevational view of the spider, and

Fig. 1'3 is a "plan view -thereof.

An input knob i is 'secured'to a shaft *2 to which is secured a cone 3,a pinion 4 and a gear 5. The cone is provided with sets of spiralthreads "6 and I which receive wires or ribbons 8 and 9 respectively,which are "wound on toor from a cylinder to. As shown in Fig. '3 theribbon =8 passes from one side of the cone to the corresponding side ofthe cylinder while the wire or ribbon 9 passes from the opposite side ofthe cone. When, therefore, the ribbon 8 is being unwound from the coneand is being wound on the cylinder, the ribbon 9 is being wound on thecone and is being unwound from the cylinder, and if these ribbons arealways maintained in a taut condition, the actuation of either the coneor the cylinder will actuate the other member.

In the above mechanism it is apparent that the most important componentis the cone, in the surface of which is cut the double helical groove orthread. One of the cables is fastened to the small end of the cone atthe point H and is wound in the thread 6 toward the large end of thecone. This ribbon or cable 8 passes from the cone to the cylinder H1 ata point depending upon the amount the cone has been rotated by the inputknob l and is now wound on the cylinder ID. The cable or ribbon 8 isattached to the cylinder at the point 12. The other cable 9 is attachedto the large end of the cone at the point l3 and as the cable 8 is woundon the cylinder in as above described, the ribbon 8 will be unwound fromthe cylinder onto the cone. The cable 9 is secured to the cylinder ID atthe point [4. If the cone should be rotated in a reverse direction,these motions would be reversed. When the cone is rotated with the cable8 passing therefrom near the small end of the cone, the cylinder will berotated slowly for each rotation of the cone and will be rotated morerapidly when the cable passes from the cone near the larger end thereof.The rate at which the cables move will increase uniformly from the smallend of the cone to the large end.

Obviously if an indicator were directly actuated by the cylinder and ifthe cone tapered down to an absolute zero value, the indicator wouldindicate the square of a number by means of the mechanism heretoforedescribed. However, it is not possible, or at least not practicable, forthe cone to extend to a mere point. It is, therefore, necessary toprovide a differential mechanism actuated both by the cylinder and thecone shaft. The gear of the cone shaft 2 meshes with and drives the gearl5 loosely mounted on the cylinder shaft 16. The gear l5 has securedthereto a pinion I! which may be considered as a sun gear, which mesheswith and drives planet pinions l8 mounted on arbors is carried by arms'20 of a spider 2| The planet pinions 18 mesh with and drive pinions 22likewise carried by the spider on arms 23. The side faces of the pinionsI8 and 22 are out of alignment with each other so that the pinions [8mesh with the pinion I! but the pinions 22 are offset so as not to meshwith pinion I1, the pinion l1 driving the pinions l8 and the pinions [8driving the pinions 22. The pinion 24 is secured to cylinder in andmeshes with pinions 22. A cover 25 is secured to the cylinder In so asto protect a portion of the gearing. Gear 26 is secured to spider 2| byany suitable means. The gear 26 drives a gear 21 mounted on a shaft 28to which is affixed an output dial 28 having suitable calibrationsthereon starting from a zero point and which may be read relative to areference point 30.

The small pinion 4 on the cone shaft drives a gear 3| secured to a shaft32 to which is secured a pinion 33, which drives a gear 34 secured to ashaft 35, to which is secured dial 36 to indicate the number ofrevolutions of the cone shaft and cone.

4 The reading index indicator 3'! is provided ad- J'acent the said dial.

A large dial 38 is secured to the cone shaft 2 and is provided with unitcalibrations to be read relative to an index point 39. The dial 38 wouldtherefore make one revolution for each revolution of the cone, whereasthe dial 36 would make one revolution for a large number of revolutionsof the cone. This is beneficial when we wish to be exact as to thenumber of rotations to be given to the input and output in the samemanner as the minute hand on a clock in conjunction with the hour handgives a much closer reading of the time than if we simply had an hourhand. When the ribbons 8 and 9 are passing from the cone at the extremesmall end thereof or the end adjacent the point II, the indicating dials36 and 38 would be at zero. However, the zero point of the cone could beits midpoint or any other selected point, provided the gear ratio of thedifferential mechanism were accordingly modified.

The difierential mechanism may be of any standard construction havingtwo spur gears such as I! and 24 coupled by two pairs of interconnectingpinions l8 and 22 carried by spider 2|. Holding either spur gear androtating the spider rotates the other spur gear in the same direction.Holding the spider and rotating either spur gear rotates the other spurgear in the opposite direction. This enables the differential mechanismto be used as an addition or subtraction mechanism in a computer.

The cone possesses the properties of an Archimedean spiral, eachrevolution of the spiral increasing the difierential length of cable perrevolution by a constant increment. Practically, a ribbon wound onitself could be utilized so that it increases the coil diameter by twicethe ribbon thickness for each revolution. Mechanically, it is moresatisfactory to wind the ribbon in a spiral groove cut on the surface ofa cone.

If n: represents a variable such as any number which it may be desiredto square, and if c represents a constant, such as the radius of a coneat its selected zero point, the value of 2: could be obtained from theequation provided the 2x0 and the 0 could be eliminated. This would meanthat we are transferring the absolute zero point of a theoretical conetapering down to a point from the point to a part of the cone, such asthe small part thereof, which we will for illustration consider as theradius of A.". The difierential mechanism above described, which isactuated both by the cone shaft and the cylinder shaft, eliminates thevalue of 2cm provided the gearing between the gear 5 and the gear l5 hasthe same ratio as the ratio between the radius of the diameter of thezero portion of the cone is to the diameter of the cylinder. Theconstant value 0 can be eliminated by making the zero point of theoutput dial equal to the value of 0 This leaves only :1: which is thenumber which it is desired to square.

The actual operation may be better illustrated assuming that thediameter of the small portion of the cone is .5" and the diameter of thelarge portion is 1 giving a mean diameter of 1",

and that the cylinder also has a diameter of 1".

The gear ratio between the gear 5 and gear [5 would be 1:2. Let usassume that the indicators 36 and 38 are at zero. Suppose further thatit would have required 25 revolutions of the cone to have unwound "theribbons from'an actual zero point to the .5 diameter of the cone. Wewill also assume that 50 "threads are cut on the cone, thereforerequiring 50 revolutions to completely come. The actual length of thecable would therefore be wind the cable onto the cone. The mean diameter5 Both of the cables are this length of each groove on the coneprogressing from the minimum to the maximum will be pozigreater than Tocalculate the rad1al differenced between the the preceding groove meandiameter. If we 'pre- 2 3 39 gjgg z gggg fif end of the cone we may sumethat we wish to square a quantity over a p y range from 0 to 10, eachrevolution of the cone 10 (52-252 will represent of a unit. Thefollowing table 1" gives the rotations of the cone, cylinder and dif- Iferential output over the first 10 turns f t cone, but modified so as toinclude th1srad1ald1fference and will illustrate how the output isdirectly pro- The formula then becomes portional to the square of thenumber. As above 01),] mentioned, if the cone had'goned'own to a diam- Weter of 0, it would have required 25 revolutions of 50 the cone tounwind the ribbon up to the smallest portion of the actual cone. Thereason that -2d was employed is because Number of Twice the llfumbgr ofD ig iz t er ggfigg i ganglia??? g g gg ff (ii a l II On 11 I, eror s eg 1 Cylinder 2 g? 53: Dggignltal ggz With a cone of uniform taper theangularity we have this differential din eachend of the cone. of thecables leading to the cylinder causes them From the above to be undergreater tension when they leave the extremities of the cone than whenthey leave the "d2 '7866437d+'0003113:o central part Of: the cone. By,cutting one Of the Therefore 00039 34 o of t threads g igi gj' g gggfigg if zgg l 40 should therefore have .a radius at the small end 57 ci of the cone equal to .25, while the other thread fifi i ggzjpfifigmggflii iigg g g gi should have a radius at the small end of thecone 9 1 e ual to .25039634. At the lar e end of the cone of taper ofthe two threadsmay explained as tl ie first thread would have a radiusof .75 and follows in connection with Figs. 5 and 6. Let it the Secondthread would have a radius be assumed that the radius of cylinder H1 is.5" 374960366 and that the radn of the. small, mid and large By theabove described mechanism I have a 1 H e aves ownan inpu nob connec e oearnestness artists; irst. an I I H iac a my squaring mechanism may beprininch, and that the length of the cone 15 35714 cipany utilized as apart of a machine in which 31 s gigr g g ii ih l fifi digfiiggf g ai oneof the functions thereof is to perform a e 2 1 squaring operation. Itherefore do not wish to crease in radius per turn of the cone would belimit my elf to a, manual input or an indicator, intranets?new: than: rwould to that orm of an inpu and any form of an output. 512941161" g Ffat 31 i It is of course apparent that the square root .of

a 1a1 0 re resen e sai num r an rea' in der, and 2 represents thetangential distance bet square i on dials 35 and 3 By quaring tween themld portlon of the cone and'cyhnder, mechanism I therefore mean meanswhereby the the Value of h equals square of a number or the square rootof a number may be obtained. V22+"2522'015565 I realize that manychanges may be made in The taper must therefore be provided to take upthe 0f the invention as Shown vby y Of a length of 015565, The theo tial length of illustration here1n, and I therefore desire to the cable onthe lower-halfof the cone would be claim the same broadly, ept as I maylimit 252) myself in the following claims.

=5-8.-90486' Having now described my invention, :1 claim: th th d 1. Ina squarin mechanism, a rotatableconical In the above formula e .5 willbe era iusat member having a spiral groove thereon, .a rothe midpoint oftge 0011:, .25 willbe. thet radius tatable cylinder, an input means fordriving the at the small en :of he cone, and he .01 conical member andcylinder, a cable wound represents the increase in radius ,per turnofthe around the cylinder and conical member in the groove thereof, adifferential mechanism having a part actuated proportionally to therotations of the conical member, and a part actuated proportionally tothe rotations of the cylinder, and an output means actuated by saiddifferential mechanism.

2. In a squaring mechanism, an input, a conical member of uniform taperdriven thereby, a cylinder, cable means interconnecting the conicalmember and cylinder and partly wound on each, a gear train anddifierential mechanism interconnecting said conical member and cylinder,said differential mechanism including two gears driven in oppositedirections by rotation of the conical member and cylinder, and alsoincluding a spider and an output connected to the spider.

3. In a squaring mechanism, an input, conical member of uniform taper,provided with two threads extending over the efiective length of theconical member, said conical member and cylinder being rotatable onsubstantially parallel axes, a cylinder, two cables connecting saidconical member and cylinder and wound in said threads so that as onecable is wound on the cylinder from the conical member, the other cableis wound on the conical member from the cylinder, said threads havingslightly different tapers to prevent slack in the cable, a differentialgear mechanism interconnecting the conical member and the cylinder, andan output connected to the differential mechanism.

4. A function computing device comprising two rotatable members, onebeing cylindrical and having a uniform surface and the other beingtapered and having helical grooves proportioned in accordance with afunction to be computed, and a pair of flexible cables wound in saidgrooves and around said cylindrical member respectively but reverselywith respect to each other, said cables having their opposite endsconnected to the two rotatable members so that they constitute apositive driving connection therebetween.

5. A function computing device comprising two rotatable members, onebeing cylindrical and having a uniform surface and the other beingtapered and having helical grooves proportioned in accordance with afunction to be computed; the radius of one of said grooves progressivelychanging from a radius smaller than that of the other groove at thelarger portion of said tapering member to a radius larger than that ofsaid other groove at the smaller portion of the tapering member, and apair of flexible cables wound in said grooves and around saidcylindrical member, respectively, but reversely with respect to eachother, said cables having their opposite ends connected to the tworotatable members so that they constitute a positive driving connectiontherebetween.

6. A function computing device comprising two rotatable members, onebeing cylindrical and having a uniform surface and the other beingtapered and having helical grooves proportioned in accordance with afunction to be computed, a pair of flexible cables wound in said groovesand around said cylindrical member, respectively, but reversely withrespect to each other, said cables having their opposite ends connectedto the two rotatable members so that they constitute a positive drivingconnection therebetween, an output member, and an actuating mechanismfor said output member comprising differential mechanism actuated by therotation of said rotatable members.

7. In a computing device having input means and output means, means fordriving the output from the input in a desired mathematical relationshipcomprising a rotatable cylindrical member, a rotatable tapering memberwith at least one helical groove having the characteristics of anArchimedean spiral formed thereon, at least one cable attached to thecylindrical member and to the tapering member, said cable being woundaround the cylindrical member and the tapering member in the groovethereof, means for maintaining said cable taut whereby the rotation ofone of said members is at all times controlled by the rotation of theother, a differential gear mechanism operable by both the cylindricalmember and the tapering member and connecting them between the inputmeans and the output means to permit the use as a zero point of a placeon the tapering member located between the theoretical apex thereof andits base, said differential gear having a gear ratio equal to the ratiobetween the radius of the tapering member at the zero point and thediameter of the cylinder.

8. A computing device according to claim 7 in which the means formaintaining the cable taut is a second cable reversely wound withrespect to the other cable and in which a second helical groove isformed on the tapering member for operation therein of the second cable.

9. A computing device according to claim '7 which includes adifferential mechanism having planet pinion gears carried by a spider,other pinion gears carried by the spider and meshing with the planetpinion gears, a spur gear connecting with the tapering member andmeshing with the planet pinion gears, another spur gear secured to thecylinder and meshing with the other pinion gears, whereby holding eitherspur gear and rotating the spider rotates the other spur gear in thesame direction.

10. A computing device according to claim 9 in which the output means isconnected to the spider.

11. A computing device according to claim 7 which comprises tworotatable members mounted to rotate on substantially parallel axes.

12. In a function computing device having two rotatable members at leastone of which is tapered and is proportioned in conjunction with theother of said rotatable members in accordance with a. function to becomputed, a pair of flexible cables reversely wound around said membersand constituting a positive driving connection between them, means forpreventing slack in said cables which consists of separate grooves insaid tapering member each to receive one of said cables, the radius ofone of said grooves progressively changing from a radius smaller thanthat of the other groove at the larger portion of said tapering memberto a radius larger than that of said other groove at the smaller portionof the tapering member.

13. In a function computing device having an input means and an outputmeans, two rotatable members at least one of which is tapered and isproportioned in conjunction with the other of said rotatable members inaccordance with a function to be computed and means for fixing theposition of one of said rotatable members with respect to the otherincluding a cable wound around both of said members, and means forsecuring at a position of finite radius on said tapering member theeffective operation which would be given by a position of zero radiuswhich consists of a difierential gear system interposed between saidinput means and output means and 9 having driving connection to saidrotatable Number members. 2,065,484 LEWIS WILLIAM IMM. 2,179,8412,194,477 REFERENCES CITED 5 2,295,997 The following references are ofrecord in the 2,349,118 file of this patent:

UNITED STATES PATENTS Number Number Name Date 10 1,847 1,581,697Stockemann Apr. 20, 1926 140, 1,920,024 Stehli July 25, 1933 339,638

Name Date Werder Dec. 22, 1936 Cassidy Nov. 14, 1939 Maxson et a1. Mar.26, 1940 Maxson et al Sept. 15, 1942 Simpson May 16, 1944;

FOREIGN PATENTS Country Date Great Britain Jan. 23, 1914 Great BritainApr. 28, 1921 Great Britain Dec. 12, 1930

